Compact cactus representations of all non-trivial min-cuts

Recently, Kawarabayashi and Thorup presented the first deterministic edge-connectivity recognition algorithm in near-linear time. A crucial step in their algorithm uses the existence of vertex subsets of a simple graph G on n vertices whose contractions leave a multigraph with Õ(n∕δ) vertices and Õ(n) edges that preserves all non-trivial min-cuts of G, where δ is the minimum degree of G and Õ hides logarithmic factors. We present a simple argument that improves this contraction-based sparsifier by eliminating the poly-logarithmic factors, that is, we show a contraction-based sparsification that leaves O(n∕δ) vertices and O(n) edges, preserves all non-trivial min-cuts and can be computed in near-linear time Õ(m), where m is the number of edges of G. We also obtain that every simple graph has O((n∕δ)²) non-trivial min-cuts. Our approach allows to represent all non-trivial min-cuts of a graph by a cactus representation, whose cactus graph has O(n∕δ) vertices. Moreover, this cactus representation can be derived directly from the standard cactus representation of all min-cuts in linear time. We apply this compact structure to show that all min-cuts can be explicitly listed in Õ(m)+O(n²∕δ) time for every simple graph, which improves the previous best time bound O(nm) given by Gusfield and Naor.